Fission — chain reaction, critical mass — Hinglish notes

2.3.23 · Physics › Modern Physics

1. What is Fission?

Ek typical reaction: $_{92}^{235} U +_{0}^{1} n ⟶_{56}^{141} Ba +_{36}^{92} Kr + 3_{0}^{1} n + Q$

Deriving the energy released (from first principles)

Fragments iron peak ( $8.8$ MeV) par nahi hain balki $A \approx 90 - 140$ region mein hain, jahan BE/A $\approx 8.5$ MeV hai. Toh:

Energy per nucleon gained $\approx (8.5 - 7.6) MeV \approx 0.9 MeV$ . Total nucleons $\approx 235$ .

$Q \approx 235 \times 0.9 MeV \approx 200 MeV$

Yeh step kyun? ~235 nucleons mein se har ek approximately 0.9 MeV zyada tightly bound ho jaata hai (fragments ki actual BE/A $8.5$ MeV use karke, iron peak nahi), aur "zyada tightly bound" matlab energy di gayi. Per-nucleon gain ko nucleons ki sankhya se multiply karo.

2. The Chain Reaction

Deriving exponential growth

Maano $N_{0}$ = generation 0 mein neutrons. $n$ generations ke baad: $N_{n} = N_{0} k^{n}$

Yeh step kyun? Har generation count ko $k$ se multiply karti hai ( $k$ ki definition). Repeated multiplication = ek power. Agar $k > 1$ yeh explode karta hai; agar $k < 1$ yeh decay karta hai.

Real time mein, generation time $τ$ ke saath (fissions ke beech ka waqt, fast neutrons ke liye $\sim 1 0^{- 8}$ s): $N (t) = N_{0} k^{t / τ}$

3. Critical Mass — the heart of the topic

Figure — Fission — chain reaction, critical mass

Why a sphere, and the role of density

Typical critical masses (bare sphere):

$^{235} U$ : $\approx 52$ kg
$^{239} Pu$ : $\approx 10$ kg

Ek neutron reflector (tamper) bhaagte huye neutrons ko wapas bounce karta hai, critical mass ko kam karta hai.

4. Reactor vs Bomb — the moderator twist

	Reactor	Bomb
Goal	$k = 1$ (steady)	$k > 1$ (rapid)
Neutrons used	slow (moderated)	fast
Fuel enrichment	~3% $^{235}$ U	>90% $^{235}$ U
Control	control rods	koi nahi — supercritical assembly chahiye

5. Worked Examples

Worked example Example 3 — Surface/volume argument numerically

Aao ek clean toy model banate hain. Production rate $\propto R^{3}$ (volume) aur dominant loss jo chhote lumps ko todti hai — surface leakage — $\propto R^{2}$ . Define karo $k (R) = \frac{production}{loss} = \frac{a R ^{3}}{b R ^{2}} = \frac{a}{b} R .$ Step 1: $k = 1$ set karo: $\frac{a}{b} R_{c} = 1 \Rightarrow R_{c} = \frac{b}{a}$ . Kyun? Yeh ek finite critical radius deta hai — woh threshold jahan production exactly leakage ko balance karta hai.

Step 2: $R < R_{c}$ ke liye, $k < 1$ (subcritical); $R > R_{c}$ ke liye, $k > 1$ (supercritical). Yeh kyun matter karta hai: yeh prove karta hai ki ek threshold radius/mass exist karta hai — critical mass ka existence geometric hai, koi magic nahi. (Reality mein ek volume-proportional absorption term bhi hota hai; kyunki yeh production ki tarah scale karta hai, yeh bas $a$ ko rescale karta hai aur surface leakage phir bhi threshold set karti hai.)

6. Common Mistakes (Steel-man + Fix)

Recall Feynman: explain to a 12-year-old

Socho ek room mein mousetrap bhari hain, har ek ek ping-pong ball se loaded hai. Ek ball andar phenko: woh ek trap set off karta hai, jo 2–3 aur balls phenkta hai, jo aur traps set off karte hain — ek runaway burst. Yahi chain reaction hai. Ab: agar room tiny hai, toh zyaadatar balls trap hit karne se pehle khuli door se baahar ud jaati hain, aur khaas kuch nahi hota. Agar room itni badi ho, toh balls traps ko hit karti rehti hain aur poora room off ho jaata hai. "Bas itni badi" size hi critical mass hai. Uranium nuclei traps hain; neutrons ping-pong balls hain.

Flashcards

Nuclear fission kya hai?

Ek heavy nucleus ka lighter fragments mein splitting, 2–3 neutrons aur ~200 MeV energy release karta hai.

Fission energy kyun release karta hai?

Fragments mein uranium (~7.6 MeV) se zyada binding energy per nucleon (~8.5 MeV) hoti hai; difference release hota hai.

^{235}

U fission se approximate energy release?

200 MeV. Multiplication factor $k$ define karo. ::: Ek generation ke neutrons ka pichli generation ke neutrons se ratio; $k = N_{n} / N_{n - 1}$ . $k < 1$ , $k = 1$ , $k > 1$ ka kya matlab hai? ::: Subcritical (khatam ho jaata hai), critical (steady self-sustaining), supercritical (exponentially badhta hai). $n$ generations ke baad neutron count ka formula? ::: $N_{n} = N_{0} k^{n}$ . Critical mass define karo. ::: Fissile material ki minimum mass jo self-sustaining chain reaction ke liye chahiye ( $k = 1$ ). Minimum (critical) mass kyun exist karta hai? ::: Production ∝ volume ( $R^{3}$ ) lekin leakage ∝ surface ( $R^{2}$ ); chhote masses ke liye leakage fraction ( $\propto 1/ R$ ) bahut bada hota hai isliye $k < 1$ . Minimum critical mass ke liye sphere optimal shape kyun hai? ::: Iska volume per surface area sabse kam hota hai → kam se kam neutron leakage. Density critical mass ko kaise affect karti hai? ::: $M_{c} \propto 1/ ρ^{2}$ ; fuel compress karne se critical mass kam hoti hai (implosion bombs ka basis). Moderator kya karta hai? ::: Fast neutrons ko thermal speeds tak slow karta hai, $^{235}$ U mein fission probability badhata hai. Control rods kya karte hain? ::: Neutrons absorb karte hain (Cd/B) reactor mein $k = 1$ hold karne ke liye. Delayed neutrons reactors ke liye crucial kyun hain? ::: Woh effective generation time (

1 0^{- 2}

–

1 0^{- 1}

s tak) badhate hain, chain reaction ko control karne ke liye itna slow banate hain.

Reactor vs bomb: required

k

Reactor

k = 1

rakhta hai (steady); bomb ko

k > 1

chahiye (rapid growth).

Connections

Binding Energy per Nucleon Curve — explain karta hai kyun fission energy release karta hai.
Nuclear Fusion — opposite process (light nuclei combine karte hain); same BE curve logic.
Mass-Energy Equivalence E=mc^2 — released $Q$ ka source.
Nuclear Reactor — controlled $k = 1$ chain ki engineering.
Radioactive Decay and Half-life — fission fragments usually radioactive hote hain.
Neutron Cross-section — slow vs fast neutron fission probability quantify karta hai.